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| Mirrors > Home > MPE Home > Th. List > undif5 | Structured version Visualization version GIF version | ||
| Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.) |
| Ref | Expression |
|---|---|
| undif5 | ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | difun2 4481 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) | |
| 2 | disjdif2 4480 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-nul 4334 |
| This theorem is referenced by: cyclnumvtx 29820 fressupp 32697 elrgspnlem4 33249 sucdifsn2 38239 |
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